Gulian A.M., Zharkov G.F. Nonequilibrium Electrons and Phonons in Superconductors

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SECTION 11.6. BRANCH IMBALANCE INITIATED BY ABSORPTION 311

where is a factor that is approximately equal to (up to the numerical

constant). From Eq. (11.77) one can see that the magnetic field renormalizes the

temperature. Making the substitution

one can restate for all the results found in Sect. 11.3 for the original temperature

T. Thus, at “temperatures” obeying the relation

the same ambiguity of arises in a thin film. As may be seen from Eqs. (11.80) and

(11.79), at the actual temperature Correspondingly, the original values

of T

and T

are also shifted. At fixed values of T

the superheated or supercooled

states may be created by varying the magnetic field. (In close analogy with the

first-order phase transition, considered in Sect. 1.2 for the Type I superconductors,

one can expect superheating and supercooling in the Type II superconductors.) In

this case the first-order phase transition in the Type II superconductors is related to

the external deviation from thermal equilibrium.

11.

. BRANCH IMBALANCE INITIATED BY ABSORPTION OF

HIGH-ENERGY QUANTA

11.

.1. Finite Curvature of the Fermi Surface

During laser pumping or high-energy particle cascade events, the branch

imbalance in superconductors is usually considered negligible. The reason is the

symmetry property

which the nonequilibrium source (5.12) [as well as its quantum generalization

(5.54)] possesses. The property (11.81) yields the solutions of the kinetic equations

of (3.63), which are symmetric over the electron and hole branches:

At the same time, there exists a mechanism that can yield the asymmetric population

of electron-hole excitations, even under the action of electromagnetic radiation.

The physical source of this asymmetry is the finite curvature of the Fermi surface.

312 CHAPTER 11. INFLUENCE OF LASER RADIATION

Namely, the density of electron levels has a large-scale dependence on the excitation

energy

This dependence was neglected in all the above calculations, since the relation

was universally adopted. One can correct the dependence (11.84) by taking into

account the relation (11.83):

Consequently, the collision integral of electrons with photons may be rewritten in

the form

where the factors were defined in Sect. 5.3.

Thus the interaction of electromagnetic radiation with the electrons in a

superconductor (as in a normal metal) “resolves” the electronlike and

holelike excitations (see Figs. 11.4 and 11.5). As a result, a photoinduced

potential must arise. We now consider this effect in more detail.

11.6.2.

Photoinduced Potential and Owen–Scalapino Model

We consider the stationary state that arises in a thin superconducting film,

which is analogous to the case considered in Chap. 5. Instead of (5.41), one deals

with (11.86), which provides, in the manner described in Sect. 5.3, the following

linearized nonequilibrium source:

We should note that the previously exploited powerful tool for the description of nonequilibrium

superconductivity, the energy-integrated Green's function technique, is not rigorously applicable

here. This technique is appropriate only for describing processes taking place in the momentum

space not far from the Fermi surface. Nevertheless, the Boltzmann-like kinetic equations could be

appropriately corrected to include the most important part of the branch imbalance effect resulting

from high-energy quanta absorption.

SECTION 11.6. BRANCH IMBALANCE INITIATED BY ABSORPTION 313

where is defined by (5.60),

and the substitution of the equilibrium function , was made in (11.87).

Owing to the presence of Eq. (11.87) implies the property:

which may be regarded as yielding a branch imbalance. Figure 11.5 illustrates the

situation clearly.

314 CHAPTER 11. INFLUENCE OF LASER RADIATION

The steady-state nonequilibrium solutions for electron-hole distribution func-

tions and in the spatially homogeneous case may be found by using the kinetic

equation

For the sake of brevity we will consider the case where the bias temperature is

absolute zero, and the electromagnetic field is “wide,” as everywhere in

this chapter. As in Sect. 11.1, we will set const, and present (11.90)

in the form [cf. (11.7)]:

In writing (11.91), we focus on the energy dependence of the distribution function

on an energy scale that is small compared with and large compared with the

nonequilibrium value of This value of may be much smaller than the

equilibrium value owing to the external pumping. Assuming this to be the case,

we set in (11.91). The presence of a superconducting state is revealed in two

aspects of the branch-imbalance solutions: they are physically allowed, and they

are stable against fluctuations.

By analogy to (11.10), let us denote

SECTION 11.6. BRANCH IMBALANCE INITIATED BY ABSORPTION 315

and assume on grounds of the phase-space volume. Proceeding as in the

derivation of (11.14), we obtain in this case:

In the absence of imbalance, the value was obtained from normalization

condition (11.3). In the case of imbalance

where Consequently,

Actually, where is the gauge-invariant potential (8.6), repre-

senting the difference between the potentials of the normal and superfluid compo-

nents of the electron liquid. On somewhat different grounds, this potential was first

introduced by Owen and Scalapino

(the so-called -model). Despite the exist-

ence of a nonzero potential following laser pumping of superconductors, it has

not yet been experimentally confirmed. A while ago the pioneering work of

Testardi

initiated interest in the states generated by the influence of electromag-

netic radiation on superconductors. This problem continues to attract the attention

of a lot of experimentalists (see, e.g., Refs. 13–17). Thus one can expect that branch

imbalance will be eventually registered and exploited in practice.

References

1. V. F. Elesin, Nonequilibrium state of superconductors with optical excitation of quasiparticles,

Sov. Phys. JETP 39(5), 862–866 (1974)

[

Zh. Eksp. i Teor. Fiz.

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pumping, Sov. Phys. JETP 44(4), 780–786 (1976)

[

Zh. Eksp. i Teor. Fiz. 71[4(10)], 1490–1502

(1976)].

3. V. Eckern, A. Schmid, M. Schmutz, and G. Schön, Stability of superconducting states out of

thermal equilibrium, J. Low Temp. Phys. 36(5/6), 643–687.

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JETP 46(1), 185–191 (1977) [Zh. Eksp. i Teor. Fiz. 73(1), 355–364 (1977)].

5. V. G. Baru and A. A. Sukhanov, New types of instabilities in nonequilibrium excitation of a

superconductor, JETP Lett. 21(4), 93–94 (1975)

[

Pis’ma v Zh. Eksp. i Tear. Fiz. 21(4), 209–212

(1975)].

6. A. G. Aronov, Paramagnetic effects in superconductors, JETP Lett. 18(6), 228–229 (1973) [Pis'ma

v Zh. Eksp. i Teor. Fiz. 18(6), 387–390 (1973)].

7. V. M. Genkin and A. P. Protogenov, Nonequilibrium states resulting from tunneling in supercon-

ductors, Sov. Phys. Solid State 18(1), 13–17 (1976) [Fiz. Tverd. Tela 18(1), 24–32 (1976)].

316 CHAPTER 11. INFLUENCE OF LASER RADIATION

8. V. F. Elesin, Phase transition in a superconductor with excess quasiparticles in an external magnetic

field, Sov. Phys. Solid State 22(10), 1808-1809 (1980) [Fiz. Tverd. Tela 22, 3097–3099 (1980)].

9. V. F. Elesin and Yu. V. Kopaev, Superconductors with excess quasiparticles, Sov. Phys. Uspekhi

24(2), 116–141 (1981) [Usp. Fiz. Nauk 133(2), 259-307 (1981)].

10. A. M. Gulian and D. Van Vechten, Electron-hole branch imbalance in superconductors initiated

by absorption of high-energy quanta, Mod. Phys. Lett. B

10(8), 329-338 (1996).

11. C. S. Owen and D. J. Scalapino, Superconducting state under the influence of external dynamic

pair breaking, Phys. Rev. Lett. 28(24), 1559-1561 (1972).

12. L. R. Testardi, Destruction of superconductivity by laser light, Phys. Rev. B 4(7), 2189–2196

(1971).

13. J. T. C. Yen and D. N. Langenberg, Light-induced superconducting weak links, Appl. Phys. Lett.

32(3), 191–192(1978).

14. Y. S. Lai, Y. Q. Liu, W. L. Cao, and C. H. Lee, Picosecond optical response of

and high superconductor films, Appl. Phys. Lett. 66(9), 1135-

1137(1995).

15. F. A. Hegmann, D. Jacobs-Perkins, C.-C. Wang, S. H. Moffat, R. A. Hughes, J. S. Preston, M.

Currie, P. M. Fauchet, T. Y. Hsiang, and R. Sobolewski, Electro-optic sampling of 1.5-ps

photoresponse signal from thin films, Appl. Phys. Lett.

(2), 285-287 (1995).

16. B. J. Feenstra, J. Schützmann, D. van der Marel, R. P. Pinaya, and M. Decroux, Nonequilibrium

superconductivity and quasiparticle dynamics studied by photoinduced activation of mm-wave

absorption, Phys. Rev. Lett. 79(24), 4890–4893 (1997).

17. D. Van Vechten, K. S. Wood, G. G. Fritz, J. S. Horwitz, G. M. Daly, J. B. Thrasher, D. M. Photiadis,

J. Ding, J. F. Pinto, M. G. Blamire, G. Burnell, A. L. Gyulamiryan, V. H. Vartanyan, R. B. Akopyan,

and A. M. Gulian, Voltage responses to optical pulses of unbiased normal and superconducting

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(10), 1415-1417 (1997).

Inverse Population Instabilities

We will refer to the inverse population state as the state with the excitation

distribution function satisfying the condition*

in a finite-energy region above the gap edge Comparing (12.1) with the

thermodynamic equilibrium function

one can call the inversely populated state a “negative temperature” state,

since at

Eq. (12.2) would satisfy (12.1).

The possibility of inverse population in superconductors challenged investiga-

tors for a long time. Despite many theoretical indications of its feasibility, and very

unusual and exciting implications (see, e.g., Refs. 2–16), this state has not yet been

realized experimentally. Depending on the frequency of external radiation, two

characteristic situations are usually considered: the case of “wide” source pumping

and the case of “narrow” source pumping.

12.1. "WIDE" PUMPING SOURCE

In wide source pumping, the distribution function is formed by the counter-

play of relaxation and recombination processes within the electron–phonon system,

so the detailed properties of may be neglected. Thus in the following section

we will focus our attention on electromagnetic pumping. The generalization of

results to tunnel injection is straightforward.

In this chapter we assume to be a real quantity and ignore the branch imbalance, setting in

all the expressions.

317

318 CHAPTER 12. INVERSE POPULATION INSTABILITIES

12.1.1. Elesin Theorem

Suppose a thin superconducting film of thickness d is deposited on a substrate,

and a monochromatic electromagnetic field is incident on the film, driving the

superconducting electron system out of equilibrium. Suppose also that the main

mechanism responsible for relaxation of the electron system toward equilibrium is

provided by inelastic collisions of electrons with phonons. The model of a “phonon

heat bath,” which was used extensively in previous chapters, simplifies the analysis.

We will consider mainly the ordinary superconductors at assuming that a

spatially homogeneous state emerges in the film under electromagnetic pumping.

The kinetic equation for electrons may be written in the form:

where is the nonequilibrium source of quasi-particles (5.21) and (5.54), and

is the electron–phonon collision integral

(4.119), which we will write in a somewhat more general form [cf. Eq. (11.34)]:

The quantity in Eq. (12.2) is the spectral function of the electron-pho-

non interaction. As noted above, we will not consider in this section inelastic

electron–electron collisions, so that the term will be ignored. To find a

proper self-consistent solution of nonlinear kinetic equations, the equation

must be coupled with Eq. (12.1).

We will consider first the isotropic model of metals with the Debye spectrum

of phonons, following Elesin.

Eq. (12.1) may be rewritten in the form:

where

SECTION 12.1. "WIDE" PUMPING SOURCE 319

and the quantities correspond to the “in” and “out” relaxation scattering

of quasi-particles, and, correspondingly, to recombination via the phonon emission.

Using (12.6), one can present as a functional fraction:

In the case of optical pumping, when the frequency greatly exceeds the value of

the source creates nonequilibrium quasi-particles in the wide range of

and inequalities

take place in the region at any practically achievable intensity of optical

pumping. This gives us grounds to rewrite (12.8) as

For further analysis, the function in (12.7) was chosen

17,18

in the

form

and the values of were studied. In all of these cases one can estimate the

values of the functionals and using for the approximation

This allows us to find the inequalities

and thus to establish (the Elesin theorem

)

from (12.8).

320 CHAPTER 12. INVERSE POPULATION INSTABILITIES

A principal difference between the kinetics of superconductors and semicon-

ductors now becomes clear: the processes of recombination in superconductors

have a single-phonon character, while in semiconductors the single-phonon proc-

esses of recombination are forbidden (because For superconductors, the

role of the semiconductor gap is performed by which does not exceed the

phonon boundary frequency in all known superconductors. Moreover, the

relation holds for traditional superconductors. At the same time, in

recently discovered high-temperature superconductors, is comparable with

In this case it is necessary to revise the earlier results.

12.1.2. "Two-Peak" Approximation for

Such an attempt was made in Ref. 19. A simple “two-peak” approximation for

the spectral function was adopted in this work:

where and are characteristic frequencies of phonons in superconductors, and

the coefficients and include the interaction constants of electrons with these

phonon modes and are also proportional to the spectral weights of these modes. For

the case

the problem permits an analytic solution, which can be presented in the form

As an analysis of this formula shows, for a narrow energy range

. The physical reasons for this result are related to the presence of

two groups of phonons, with the frequencies and obeying the relation

(12.13), which causes some peculiarities in the electron kinetics. Indeed, the

recombination takes place by the participation of phonons only. But this channel

turns out to be noneffective, because the quasi-particles in the vicinity of the gap

edge (with the energy are to recombine with the quasi-particles removed far

from the gap edge The concentration of these high-energy quasi-par-

ticles is rather low. That is why, contrary to the model used in Ref. 17, the

recombination processes are not effective in this case, and the term may be

neglected in (12.8). With respect to the interplay between factors and in (12.8),

it may be noted that at (when only spontaneous emission of phonons in a

relaxation channel may take place) outscattering of electrons from the states lower

than the level is impossible [this causes at . A